Problem 113: Non-bouncy numbers

Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for example, 134468.
Similarly if no digit is exceeded by the digit to its right it is called a decreasing number; for example, 66420.

We shall call a positive integer that is neither increasing nor decreasing a "bouncy" number; for example, 155349.

As n increases, the proportion of bouncy numbers below n increases such that there are only 12951 numbers below one-million
that are not bouncy and only 277032 non-bouncy numbers below 10^10.

How many numbers below a googol (10^100) are not bouncy?

My Algorithm

My solution is based on a dynamic programming approach:

solve the problem for one digit

solve the problem for n+1 digits by using information from n digits

All 9 single digits are not bouncy. Therefore I initialize my array increase and decrease with 1s.
Each of their entries represents how many numbers starting with a digit are not bouncy.
In increase[x][y] you find the count of increasing numbers with x digits, where the front-most digit is y.

In each iteration, increase and decrease are updated:increase[x][y] is the sum of all increase[x-1][less than or equal to y] (and the other way around for decrease, too).
All not bouncy numbers are either increasing or decreasing. I have to deduct all increasing numbers where the first digit is zero.
On top of that, numbers where all digits are identical are both increasing und decreasing and counted twice, therefore I have to subtract 10.

Changelog

Hackerrank

Difficulty

30%
Project Euler ranks this problem at 30% (out of 100%).

Hackerrank describes this problem as easy.

Note:Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.

Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.

Heatmap

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solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too

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[new]

the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.

The 310 solved problems (that's level 12) had an average difficulty of 32.6&percnt; at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of &approx;60000 in August 2017)
at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.All of my solutions can be used for any purpose and I am in no way liable for any damages caused.You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.Thanks for all their endless effort !!!

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